Order reduction for ideals

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

This lesson proves the first half of the inductive step: if marked ideals can be reduced in order in every dimension below \(n\), then every ideal on an \(n\)-dimensional smooth scheme can be reduced below its maximal order by blow-ups of that order. The proof restricts the problem to a hypersurface of maximal contact, solves it there by induction, and pushes the solution back by going up. Before restricting, the snc divisor carried along must be made transversal to the hypersurface, and this is done by the same method one divisor at a time. The uniqueness of maximal contact makes the result independent of the hypersurface, and the local constructions glue to a functor on all triples.

We use all previous lessons: Smooth blow-ups and transforms of ideals, Blow-up sequences and the main theorems, Derivative ideals under blowing up, Hypersurfaces of maximal contact, Logarithmic derivatives and going up, Uniqueness of maximal contact and Tuning of ideals.

Induction hypothesis. Throughout, \(n\ge1\), and Theorem 4.2 of Blow-up sequences and the main theorems (order reduction for marked ideals, with its functoriality for smooth morphisms and change of field, and its statement (2) on closed embeddings) holds for marked triples of dimension \(<n\), with functoriality for smooth morphisms between such triples. Functoriality in this lesson always concerns smooth morphisms between triples of dimension \(\le n\); the restricted problems then have dimension \(\le n-1\), where the hypothesis applies.

1. A transversality lemma

Lemma 1.1. Let \(Y\) be smooth, \(D_b=V(f_b)\), \(b\in B\), smooth divisors through a point \(p\) whose differentials \(df_b(p)\) are linearly independent, and \(Z\subset Y\) a smooth closed subscheme through \(p\). Let \(B'\subset B\) be the set of \(b\) with \(Z\not\subset D_b\) near \(p\). Then \(Z\) has snc with \(\{D_b\}\) at \(p\) if and only if the differentials \(d(f_b|_Z)(p)\), \(b\in B'\), are linearly independent in \(\Omega_Z\otimes\kappa(p)\).

Proof. If coordinates are adapted, with \(D_b=V(z_{c(b)})\) and \(Z=V(z_l:l\in J)\), then for \(b\in B'\) the index \(c(b)\) is not in \(J\), and the \(z_l|_Z\), \(l\notin J\), are coordinates on \(Z\); as \(f_b\) is a unit times \(z_{c(b)}\), the claim follows. Conversely, the \(f_b\) with \(b\notin B'\) lie in \(\mathcal I_Z\) and have independent differentials, so by the conormal sequence of the smooth pair \(Z\subset Y\) (as in Lemma 1.3 of the first lesson) they extend to a minimal set of generators \(\{f_b\}_{b\notin B'}\cup\{g_1,\ldots,g_t\}\) of \(\mathcal I_Z\) near \(p\), whose differentials span the conormal space of \(Z\). Adding the \(f_b\), \(b\in B'\), whose restrictions to \(Z\) are independent, and further functions whose restrictions complete them to coordinates of \(Z\), gives a coordinate system near \(p\) in which \(Z\) and every \(D_b\) are coordinate subspaces. \(\square\)

Corollary 1.2. Let \(H\) be a smooth hypersurface and \(N\) an snc divisor with \((H,N)\) snc. If a smooth centre \(Z\subset H\) has snc with \(N\), then it has snc with \((H,N)\), and \((B_ZH,\pi^{-1}_{\rm tot}N)\) is snc.

Proof. Since \(Z\subset H\), \(Z\subset N^a\) holds if and only if \(Z\subset N^a\cap H\). Apply Lemma 1.1 in \(X\) to \(N\) and \(Z\), and then in \(H\) to the divisors \(N^a\cap H\), whose differentials are independent because \((H,N)\) is snc: the restrictions to \(Z\) of their equations are the same functions. So \(Z\) has snc with \(N|_H\) in \(H\), and Logarithmic derivatives and going up, Lemma 4.2 gives the claims. \(\square\)

2. Moving the high-order locus off a divisor

Lemma 2.1. For every \(m\ge1\) and \(j\ge1\) there is a blow-up sequence functor \(\mathcal B^{\rm dis}_{m,j}\) of order \(m\), defined on triples \((X,\mathcal I,E)\) with \(\dim X=n\), \(\operatorname{maxord}\mathcal I\le m\) and \(E=(E^1,\ldots,E^s)\), \(s\ge j\), such that at its end

\[ \operatorname{cosupp}(\mathcal I_r,m)\cap\Pi^{-1}_*E^j=\emptyset, \]

and which commutes with smooth morphisms and change of field. Moreover, if \(m=1\), \(E^j\) is nonempty with inclusion \(\tau:E^j\to X\), and \(\mathcal O_X/\mathcal I=\tau_*(\mathcal O_{E^j}/\mathcal J)\) for an ideal \(\mathcal J\) nonzero on every component of \(E^j\), then

\[ \mathcal B^{\rm dis}_{1,j}(X,\mathcal I,E)=\tau_*\mathcal B^{\rm mord}_1\bigl(E^j,\mathcal J,1,(E-E^j)|_{E^j}\bigr). \]

The same construction applies with the index \(j=0\) when \(E\) is written \((E^0,E^1,\ldots,E^s)\).

Proof. If \(\operatorname{maxord}\mathcal I<m\), the sequence is empty. Assume \(\operatorname{maxord}\mathcal I=m\), and let \(W=W(\mathcal I)\) be the tuned ideal, of maximal order \(w=s(m)\), D-balanced and MC-invariant (Tuning of ideals, Corollary 2.2). Blow-up sequences of order \(m\) for \((X,\mathcal I,E)\) are the same as blow-up sequences of order \(w\) for \((X,W,E)\), with the same end cosupports: a closed point is a centre of order \(m\) for \(\mathcal I_r\) exactly when it is one of order \(w\) for \(W_r\), by the same corollary, and cosupports are the closures of their closed points.

Step 1. Let \(Z_{-1}\) be the union of the components of \(E^j\) contained in \(\operatorname{cosupp}(W,w)\). Blow it up; this is a trivial blow-up of order \(w\) whose centre has snc with \(E\), giving \((X_0,W_0,E_0)\) with \(X_0=X\), \(W_0=\mathcal O_X(wZ_{-1})W\), and \(E_0\) obtained from \(E\) by removing \(Z_{-1}\) from \(E^j\) and appending \(Z_{-1}\) as the last member. Near a component \(C\) of \(Z_{-1}\), \(W\) has order exactly \(w=\operatorname{maxord}W\) at every point of \(C\), so \(W\subset\mathcal O(-wC)\) by Smooth blow-ups and transforms of ideals, Lemma 4.1. Write \(W=\mathcal O(-wC)W'\) near \(C\). Orders add, so at every point \(p\in C\), \(\operatorname{ord}_pW'=\operatorname{ord}_pW-w=0\); hence \(W'=\mathcal O\) and \(W=\mathcal O(-wC)\) near \(C\). Hence \(W_0=\mathcal O\) near \(Z_{-1}\) and \(W_0=W\) elsewhere, so \((W_0,w)\) is still D-balanced in the sense of Definition 1.1 of that lesson, and \(\operatorname{cosupp}(W_0,w)\) contains no component of \(S:=E^j_0=E^j\setminus Z_{-1}\).

Step 2. Consider the marked triple \((S,W_0|_S,w,E_S)\) of dimension \(n-1\), where \(E_S\) is the restriction to \(S\) of the members of \(E_0\) other than \(E^j_0\) (the last member \(Z_{-1}\) does not meet \(S\) and restricts to the empty divisor). It is a marked triple: \(W_0|_S\) is nonzero on every component because \(\operatorname{cosupp}(W_0,w)=V(W_0)\) contains no component of \(S\), and \(E_S\) is snc because \(E\) is. Let \(\mathbf B^S=\mathcal B^{\rm mord}_w(S,W_0|_S,w,E_S)\), given by the induction hypothesis, and push it forward to \(X_0\). By going up (Logarithmic derivatives and going up, Theorem 4.1) it is a sequence of order \(\ge w\), hence of order \(w\), for \((X_0,W_0)\), whose transforms restrict to those of \(\mathbf B^S\). By Lemma 4.2 of that lesson, applied at every step with the hypersurface \(S_i\) and the divisor formed by the other members, each centre has snc with the whole total transform \(E_i\). Define \(\mathcal B^{\rm dis}_{m,j}(X,\mathcal I,E)\) as the blow-up of \(Z_{-1}\) followed by this push-forward.

End result. By Logarithmic derivatives and going up, Corollary 3.2, the term \(j=0\) gives \(S_r\cap\operatorname{cosupp}(W_r,w)\subset\operatorname{cosupp}((\Pi^S)^{-1}_*(W_0|_S,w),w)\), and the right side is empty because \(\mathbf B^S\) reduces the order below \(w\). The strict transform of \(E^j\) is \(S_r\), up to the components \(Z_{-1}\) whose strict transforms are empty. Hence \(\operatorname{cosupp}(\mathcal I_r,m)\cap\Pi^{-1}_*E^j=\emptyset\).

Functoriality. Let \(h:Y\to X\) be smooth. If \(h(Y)\) misses \(\operatorname{cosupp}(\mathcal I,m)\), the sequence on \(Y\) is empty, and so is every pulled-back centre, because all centres lie over \(\operatorname{cosupp}(\mathcal I,m)\). Otherwise \(W(h^*\mathcal I)=h^*W\) (Tuning of ideals, Definition 2.3). A component \(C'\) of \(h^{-1}(E^j)\) lies in \(h^{-1}\operatorname{cosupp}(W,w)\) if and only if the component \(C\) of \(E^j\) under it lies in \(\operatorname{cosupp}(W,w)\): the image of \(C'\) is open and dense in \(C\), because \(h^{-1}(C)\to C\) is smooth, and the cosupport is closed. So the first centre pulls back to the first centre. Restriction to \(S\) commutes with pull-back along the smooth map \(h^{-1}(S)\to S\), and \(\mathcal B^{\rm mord}_w\) commutes with smooth morphisms by the induction hypothesis; push-forward commutes with pull-back. Change of field is the same argument.

Closed embeddings. In the last statement, \(\mathcal I\) contains the local equations of \(E^j\), so \(\operatorname{maxord}\mathcal I=1\), \(W=\mathcal I\) and \(w=1\). A component of \(E^j\) inside \(\operatorname{cosupp}(\mathcal I,1)=V(\mathcal I)\) would be a component on which \(\mathcal J\) vanishes, so \(Z_{-1}=\emptyset\) and \(W_0=\mathcal I\). Finally \(\mathcal I|_{E^j}=\mathcal J\), because \(\mathcal I\supset\mathcal I_{E^j}\) and \(\mathcal I/\mathcal I_{E^j}=\mathcal J\). \(\square\)

3. Order reduction for ideals in dimension \(n\)

Theorem 3.1. Under the induction hypothesis, for every \(m\ge1\) there is a blow-up sequence functor \(\mathcal B^{\rm ord}_m\) of order \(m\), defined on triples \((X,\mathcal I,E)\) with \(\dim X=n\) and \(\operatorname{maxord}\mathcal I\le m\), such that \(\operatorname{maxord}\mathcal I_r<m\) at its end, and which commutes with smooth morphisms and change of field. Moreover, if \(\tau:Y\to X\) is a smooth hypersurface and \(\mathcal O_X/\mathcal I=\tau_*(\mathcal O_Y/\mathcal J)\) with \(\mathcal J\) nonzero on every component of \(Y\), then \(\operatorname{maxord}\mathcal I=1\) and

\[ \mathcal B^{\rm ord}_1(X,\mathcal I,\emptyset)=\tau_*\mathcal B^{\rm mord}_1(Y,\mathcal J,1,\emptyset). \]

Proof. If \(\operatorname{maxord}\mathcal I<m\) the sequence is empty, so assume \(\operatorname{maxord}\mathcal I=m\). Let \(\mathcal{LT}\) be the class of such triples on which there is an MC-hypersurface \(H\) for \(\mathcal I\) (Definition 2.2 of Hypersurfaces of maximal contact), together with the triples with \(\operatorname{maxord}\mathcal I<m\). By Theorem 2.3(1) there, every triple is covered by open subsets in \(\mathcal{LT}\). The class is closed under open subsets and disjoint unions, and under pull-back by smooth morphisms, by Theorem 2.3(5) there. We construct the functor on \(\mathcal{LT}\) and then glue.

Step A: making the old divisors disjoint from the cosupport. Write \(E=(E^1,\ldots,E^s)\). For \(i=1,\ldots,s\) in turn, apply \(\mathcal B^{\rm dis}_{m,i}\) to the current triple; the index \(i\) refers to the strict transform of the original member \(E^i\), since new members are appended at the end. Stop as soon as the maximal order drops below \(m\). This uses no choice. At the end, \((X',\mathcal I',E')\) satisfies \(\operatorname{cosupp}(\mathcal I',m)\cap\Pi^{-1}_*E^i=\emptyset\) for all \(i\le s\): the step for \(i\) achieves it for \(E^i\), and later steps preserve it, because their centres lie in the cosupport and the cosupport of a transform lies over the cosupport and the new exceptional divisor, while the strict transform of \(E^i\) meets the new exceptional divisor only over \(E^i\cap Z\), which is empty.

Let \(H\) be an MC-hypersurface for \(\mathcal I\) and \(H'\) its strict transform. Every centre in Step A lies in the strict transform of \(H\) by Theorem 2.3(2) of Hypersurfaces of maximal contact, and if \(\operatorname{maxord}\mathcal I'=m\), then \(H'\) is an MC-hypersurface for \(\mathcal I'\) containing \(\operatorname{cosupp}(\mathcal I',m)\), by Theorem 2.3(4). Let \(N\) be the ordered divisor of the members created in Step A. By induction over the blow-ups of Step A, using Corollary 1.2 at each step (the centre has snc with the current total transform, hence with the new members), \((H',N)\) is snc. Near \(\operatorname{cosupp}(\mathcal I',m)\) the old members are absent, so on an open neighbourhood \(U\) of \(\operatorname{cosupp}(\mathcal I',m)\) the ordered divisor \((H',E')\) is snc.

Step B: restricting to the hypersurface. On \(U\), let \(W'=W(\mathcal I')\), of maximal order \(w\), D-balanced and MC-invariant, with the same order-\(m\) sequences as \(\mathcal I'\) (Corollary 2.2 of the tuning lesson). Its MC-hypersurfaces are those of \(\mathcal I'\), because \(MC(W')=MC(\mathcal I')\) (Proposition 1.3(4) there). Apply \(\mathcal B^{\rm dis}_{w,0}\) of Lemma 2.1 to \((U,W',(H',E'))\), with \(H'\) as the member of index \(0\). Its centres lie in the cosupport, which is closed in \(X'\) and contained in \(U\), so the sequence extends uniquely to \(X'\). At its end the cosupport of the transform of \(W'\), which equals that of \(\mathcal I'\) by tuning, is disjoint from the strict transform of \(H'\). But the strict transform of \(H'\) contains this cosupport by Theorem 2.3(4) of the maximal contact lesson, as long as the maximal order is still \(m\). So the cosupport is empty: \(\operatorname{maxord}<m\). Define \(\mathcal B^{\rm loc}(X,\mathcal I,E)\) as Step A followed by Step B.

Independence of \(H\). Step A does not involve \(H\). Let \(H_1,H_2\) be two MC-hypersurfaces for \(\mathcal I\), with strict transforms \(H_1',H_2'\) after Step A. Both are MC-hypersurfaces for \(\mathcal I'\), hence for \(W'\), and \((H_1',E')\), \((H_2',E')\) are snc on a common neighbourhood \(U\) of the cosupport. By Uniqueness of maximal contact, Theorem 3.3, applied to the MC-invariant ideal \(W'\), they are étale equivalent with respect to \((U,W',E')\). The functor \(\mathcal C(U,W',(H,E'))=\mathcal B^{\rm dis}_{w,0}(U,W',(H,E'))\) commutes with étale morphisms by Lemma 2.1 and produces sequences of order \(w\) for \((U,W',E')\). By Corollary 4.2 of the uniqueness lesson, the two choices give the same sequence. So \(\mathcal B^{\rm loc}\) is well defined on \(\mathcal{LT}\).

Functoriality on \(\mathcal{LT}\). Let \(h:Y\to X\) be smooth, with both triples in \(\mathcal{LT}\). Step A commutes with \(h\) by Lemma 2.1. If \(h(Y)\) meets the cosupport, \(h^{-1}(H)\) is an MC-hypersurface for \(h^*\mathcal I\), its strict transform is \(h'^{-1}(H')\) for the induced \(h':Y'\to X'\), tuning commutes with \(h'\), and Lemma 2.1 gives \(h'^*\) of Step B on \(X\) equal to Step B on \(Y\) computed with \(h^{-1}(H)\). By independence of the choice, this is Step B on \(Y\). If \(h(Y)\) misses the cosupport, both sides are empty. Change of field is the same.

Gluing. By the gluing theorem (Blow-up sequences and the main theorems, Theorem 5.1), applied with \(\mathcal{GT}\) the class of all triples of dimension \(n\) with \(\operatorname{maxord}\le m\), \(\mathcal B^{\rm loc}\) extends uniquely to a functor \(\mathcal B^{\rm ord}_m\) on \(\mathcal{GT}\) commuting with smooth morphisms and change of field. At its end the maximal order is \(<m\), because this holds on the pieces of a cover in \(\mathcal{LT}\).

Closed embeddings. In the last statement \(\mathcal I\) contains the local equations of \(Y\), so \(\operatorname{maxord}\mathcal I=1\) and \(MC(\mathcal I)=\mathcal I\). Hence \(Y\) is a global MC-hypersurface, and the triple lies in \(\mathcal{LT}\). Step A is empty since \(E=\emptyset\), \(W(\mathcal I)=\mathcal I\), and we may take \(H=Y\) and \(U=X\). By the last statement of Lemma 2.1 (with \(j=0\)), Step B is \(\tau_*\mathcal B^{\rm mord}_1(Y,\mathcal J,1,\emptyset)\). By independence of the choice, this is \(\mathcal B^{\rm ord}_1(X,\mathcal I,\emptyset)\). \(\square\)

4. An example

Example 4.1. Let \(\mathcal I=(x^2+y^3)\) on \(\mathbf A^2\) with \(E=\emptyset\) and \(m=2\). Step A is empty. The hypersurface \(H=V(x)\) is an MC-hypersurface, \(W=W(\mathcal I)\) has maximal order \(4\) and \(W|_H=(y^6)\) (Example 3.1 of the tuning lesson). In Lemma 2.1 with \(j=0\), \(Z_{-1}=\emptyset\), because \(H\) is not contained in \(\operatorname{cosupp}(W,4)\), which is the origin. The marked triple \((H,(y^6),4,\emptyset)\) has dimension one. Its order reduction for marked ideals (Order reduction for marked ideals) blows up the origin of \(H\) once, a trivial blow-up, after which the marked transform is \(y^{-4}(y^6)=(y^2)\), of order \(<4\). Pushed forward, this is the blow-up of the origin of \(\mathbf A^2\), after which \(\mathcal I\) has maximal order \(1\).

5. Exercises

Exercise 5.1. Run Theorem 3.1 for \(\mathcal I=(x^2+y^3)\) on \(\mathbf A^2\), \(m=2\), with \(E=(V(y))\).

Solution. Step A applies Lemma 2.1 with \(j=1\). The tuned ideal \(W\) has maximal order \(4\) and cosupport the origin, so \(Z_{-1}=\emptyset\) and \(S=V(y)\cong\mathbf A^1_x\). Since \(\mathcal I|_S=(x^2)\) and \(D(\mathcal I)|_S=(x,y^2)|_S=(x)\), \(W|_S=(x^4)+(x^2)(x)^2+(x)^4=(x^4)\). Order reduction for \((S,(x^4),4,\emptyset)\) blows up the origin of \(S\) once. Pushed forward, Step A blows up the origin of \(\mathbf A^2\). Afterwards \(\mathcal I\) has transform \((x_1^2+y)\) on the chart \(x=x_1y\) and \((1+y_1^3x)\) on the chart \(y=y_1x\), of maximal order \(1\). So the construction stops after Step A, and the final divisor is the strict transform of \(V(y)\) followed by the exceptional curve.

Exercise 5.2. Let \(\mathcal I=(x^2)\) on \(\mathbf A^2\), \(m=2\), \(E=(V(x))\). Run Lemma 2.1 with \(j=1\).

Solution. \(W=W(\mathcal I)=(x^4)\) with \(w=4\). The member \(E^1=V(x)\) lies in \(\operatorname{cosupp}(W,4)\), so Step 1 blows it up trivially: \(W_0=\mathcal O(4E^1)(x^4)=\mathcal O\). Then \(S=\emptyset\), Step 2 is empty, and the cosupport is already empty. The output is the single trivial blow-up of \(V(x)\), and \(E_0=(\emptyset,V(x))\).

Exercise 5.3. Verify Lemma 1.1 for \(Y=\mathbf A^3\), \(D_1=V(y)\), and \(Z=V(x,y-z)\).

Solution. \(Z\not\subset D_1\), and \(f_1|_Z=y|_Z\), which is a coordinate on the line \(Z\) (parametrized by \(y=z\)), so its differential is nonzero. Indeed in the coordinates \((x,y,y-z)\), \(D_1=V(y)\) and \(Z=V(x,y-z)\) are coordinate subspaces.

References